BARCELONA TOPOLOGY WORKSHOP 2006

Author: Joachim Kock, Universitat Autonoma de Barcelona 
    
Title: Commutativity in double semigroups and symmetry in two-fold 
  monoidal categories

Abstract: A concrete computation - twelve slidings with sixteen 
  tiles - reveals that certain commutativity phenomena occur in every
  double semigroup.  This can be seen as a sort of Eckmann-Hilton 
  argument, but it does not use units.  The result implies in 
  particular that all cancellative double semigroups and all inverse 
  double semigroups are commutative.  Stepping up one dimension, the 
  result is used to prove that all strictly associative two-fold 
  monoidal categories are symmetric.  In particular, strictly 
  associative one-object, one-arrow 3-groupoids (but still with weak 
  units) cannot realise all simply-connected homotopy 3-types.